Inequalities in modulus & exponents : Data Sufficiency

Inequalities, modulus and exponents go together with data sufficiency. When will the comparison asked in this question hold good - what happens if one of the numbers is negative? Unless stated explicity, do not assume numbers to take only positive values or only integer values.

Directions

  • DS
  • This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether:

    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
    C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
    D. EACH statement ALONE is sufficient to answer the question asked.
    E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

 

Question

If m and n are non-zero integers, is mn > nn?
Statement 1: |m| = n
Statement 2: m < n

 

  • Correct Answer
    Choice A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Explanatory Answer

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Detailed Solution

Answer the following questions before you read the statements.

1. What kind of an answer will the question fetch?

An "Is" question will fetch an "Yes" or a "No" as an answer.

The data provided in the statements will be considered sufficient if the question is answered with a conclusive Yes or a conclusive No.

2. When is the answer an "Yes"?

If mn > nn, the answer to the question is a conclusive Yes.

3. When is the answer a "No"?

If mn ≤ nn, the answer to the question is a conclusive No.

Note: When mn = nn, the answer is No.

It pays rich dividend to note down the answer to questions 2 and 3 mentioned above in your scratch paper while solving DS questions.

4. What values can m and n take?

From the information available from the question stem, both m and n can take only integer values. So, we need not worry about values such as 0.5 or 1.2. However, both m and n can be either positive or negative. Neither can be 0.

Shortlisting range of values that the numbers can take will save you a lot of potential trouble while evaluating the statements.

Now let us evaluate each statement independently.

Statement 1: |m| = n

We can infer the following information about m and n from the statement.

  1. The modulus of a number is always positive. n = |m|. Hence, n is positive
  2. m can take either positive or negative values.

Look for counter examples

A good approach to solving such questions is to figure out two counter examples. i.e., find two values for m and n that satisfy the condition stated in the statement - one set of values satisfying the inequality and the second set negating it.

If you manage to find two such sets of values, then you can conclude that the data given in statement 1 is not sufficient.

However, if the values that you have taken give a uniform answer do not rush to the conclusion that staement 1 is sufficient. Dig a little deeper to reason out why it happens - many a times when you try to find the reason, you will stumble upon the counter example if one existed

Example 1: Let m = -3 and n = 3. |m| = n holds good.
(-3)3 < 33. So, the answer to the question is NO.

Example 2: Let m = -2 and n = 2. |m| = n still holds good.
(-2)2 = 22. So, the answer to the question is NO.

We seem to have got a uniform answer with both the values. We had not assumed m to be positive in both the examples. When m is positive, LHS = RHS. Hence, the answer will still be a NO.
Let us look for the rationale. When a negative number is raised to an odd power, the result is negative. So, LHS < RHS. Answer is NO.
When a negative number is raised to an even power, the result is positive. So, LHS = RHS. It is still not greater. So, the answer will still be NO.
Because m and n are integers, only things to check for is what happens when m is negative when the numbers are odd and what happens when they are even. There is no other possiblility.

Because we have got a uniform answer using information in statement 1, statement 1 is Sufficient.

Therefore, the answer will either be choice A or choice D.

Statement 2: m < n

We need to check if we get a conclusive Yes or No using this statement to determine whether statement 2 alone is sufficient

Let us look for counter examples

Example 1: Let m = 2 and n = 3
23 < 33. So, the answer to the question is NO.

Example 2: Let m = -3 and n = 2
(-3)2 > 22. So, the answer to the question is YES.

As we are not getting a consistent answer to the question from statement 2, statement 2 alone is NOT Sufficient.

The answer

Statement 1 alone is sufficient to answer the question; statement 2 is not sufficient to answer the question.

Choice A is the correct answer.